What are the asymptotes and removable discontinuities, if any, of #f(x)= (x^2+x-12)/(x^2-4)#?
1 Answer
Explanation:
#"factorise numerator/denominator"#
#f(x)=((x+4)(x-3))/((x-2)(x+2))#
#"there are no common factors on numerator/denominator"#
#"hence there are no removable discontinuities"# The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.
#"solve "(x-2)(x+2)=0#
#rArrx=+-2" are the asymptotes"#
#"horizontal asymptotes occur as"#
#lim_(xto+-oo),f(x)toc" ( a constant)"# Divide terms on numerator/denominator by the highest power of x that is
#x^2#
#f(x)=(x^2/x^2+x/x^2-12/x^2)/(x^2/x^2-4/x^2)=(1+1/x-12/x^2)/(1-4/x^2)#
#"as "xto+-oo,f(x)to(1+0-0)/(1-0)#
#rArry=1" is the asymptote"#
graph{(x^2+x-12)/(x^2-4) [-20, 20, -10, 10]}